To get dimension 4, I think the norm \begin{equation} \|a\| = \max_{\{i,j\}} |a_i-a_j| \vee \|a\|_\infty, \end{equation} where $a=(a_1,a_2,a_3,a_4)$, works. Again $X$ samples the unit vector basis uniformly.
Bill Johnson
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